Quantitative estimates for uniformly-rotating vortex patches

TL;DR

This paper provides quantitative estimates for uniformly-rotating vortex patches, showing how their shape and angular velocity relate, especially for small angular velocities and high symmetry cases.

Contribution

It establishes bounds on the distance of the outermost point from the center and characterizes the angular velocity for symmetric patches, advancing understanding of vortex patch dynamics.

Findings

Outermost point distance scales as ()^{-1/2} for small .

Angular velocity approaches 1/2 for large symmetry order m.

Provides estimates on the boundary's polar graph in the -norm.

Abstract

In this paper, we derive some quantitative estimates for uniformly-rotating vortex patches. We prove that if a non-radial simply-connected patch $D$ is uniformly-rotating with small angular velocity $0 < \Omega \ll 1$, then the outmost point of the patch must be far from the center of rotation, with distance at least of order $\Omega^{-1/2}$. For $m$-fold symmetric simply-connected rotating patches, we show that their angular velocity must be close to $\frac{1}{2}$ for $m\gg 1$ with the difference at most $O(1/m)$, and also obtain estimates on $L^{\infty}$ norm of the polar graph which parametrizes the boundary.